Normalization of Typable Terms by Superdevelopments

  • Zurab Khasidashvili
  • Adolfo Piperno
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1584)


We define a class of hyperbalancedλ-terms by imposing syntactic constraints on the construction of λ-terms, and show that such terms are strongly normalizable. Furthermore, we show that for any hyperbalanced term, the total number of superdevelopments needed to compute its normal form can be statically determined at the beginning of reduction. To obtain the latter result, we develop an algorithm that, in a hyperbalanced term M, statically detects all inessential (or unneeded) subterms which can be replaced by fresh variables without effecting the normal form of M; that is, full garbage collection can be performed before starting the reduction. Finally, we show that, modulo a restricted η-expansion, all simply typable λ-terms are hyperbalanced, implying importance of the class of hyperbalanced terms.


Normal Form Induction Assumption Garbage Collection Typable Term Strong Normalization 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Asperti, A., Danos, V., Laneve, C., Regnier, L.: Paths in the lambda-calculus. In: Proceedings of the Symposium on Logic in Computer Science (LICS). IEEE Computer Society Press, Los Alamitos (1994)Google Scholar
  2. 2.
    Asperti, A., Mairson, H.G.: Parallel beta reduction is not elementary recursive. In: Proc. of ACM Symposium on Principles of Programming Languages, POPL (1998)Google Scholar
  3. 3.
    van Bakel, S.: Intersection Type Disciplines in Lambda Calculus and Applicative Term Rewriting Systems. PhD thesis, Matematisch Centrum Amsterdam (1993)Google Scholar
  4. 4.
    Barendregt, H.: The Lambda Calculus. In: Its syntax and Semantics (revised edition). North Holland, Amsterdam (1984)Google Scholar
  5. 5.
    Barendregt, H.: Lambda Calculi with Types. In: Handbook of Logic in Computer Science, vol. 2. Oxford University Press, Oxford (1992)Google Scholar
  6. 6.
    Barendregt, H.P., Kennaway, J.R., Klop, J.W., Sleep, M.R.: Needed Reduction and spine strategies for the lambda calculus. Information and Computation 75(3), 191–231 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Coppo, M., Dezani-Ciancaglini, M.: An Extension of the Basic Functionality Theory for the λ-Calculus. Notre Dame J. of Formal Logic 21(4), 685–693 (1980)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Girard, J.-Y.: Une extension de l’interprétation de Gödel à l’analyse, et son application à l’elimination des coupures dans l’analyse et la théorie des types. In: Procs of 2nd Scandinavian Logic Symposium, North-Holland, Amsterdam (1971)Google Scholar
  9. 9.
    Glauert, J.R.W., Khasidashvili, Z.: Relative Normalization in Orthogonal Expression Reduction Systems. In: Lindenstrauss, N., Dershowitz, N. (eds.) CTRS 1994. LNCS, vol. 968, pp. 144–165. Springer, Heidelberg (1995)Google Scholar
  10. 10.
    Huet, G., Lévy, J.-J.: Computations in Orthogonal Rewriting Systems. In: Robinson, A., Lassez, J.-L., Plotkin, G. (eds.) Computational Logic. MIT Press, Cambridge (1991)Google Scholar
  11. 11.
    Khasidashvili, Z.: β-reductions and β-developments of λ-terms with the least number of steps. In: Martin-Löf, P., Mints, G. (eds.) COLOG 1988. LNCS, vol. 417, pp. 105–111. Springer, Heidelberg (1990)Google Scholar
  12. 12.
    Khasidashvili, Z.: On higher order recursive program schemes. In: Tison, S. (ed.) CAAP 1994. LNCS, vol. 787, pp. 172–186. Springer, Heidelberg (1994)CrossRefGoogle Scholar
  13. 13.
    Khasidashvili, Z.: On Longest Perpetual Reductions in Orthogonal Expression Reduction Systems (Submitted)Google Scholar
  14. 14.
    Klop, J.W., van Oostrom, V., van Raamsdonk, F.: Combinatory reduction systems: introduction and survey. Theoretical Computer Science 121, 279–308 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Klop, J.W.: Combinatory Reduction Systems. PhD thesis, Matematisch Centrum Amsterdam (1980)Google Scholar
  16. 16.
    Lévy, J.-J.: An algebraic interpretation of the λβ κ-calculus and a labelled λ- calculus. Theoretical Computer Science 2, 97–114 (1976)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Lévy, J.-J.: Réductions correctes et optimales dans le λ-calcul. PhD thesis, Univerité Paris 7 (1978)Google Scholar
  18. 18.
    Piperno, A., Ronchi della Rocca, S.: Type inference and extensionality. In: Proceedings of the Symposium on Logic in Computer Science (LICS). IEEE Computer Society Press, Los Alamitos (1994)Google Scholar
  19. 19.
    Piperno, A.: Normalization and extensionality. In: Proceedings of the Symposium on Logic in Computer Science (LICS). IEEE Computer Society Press, Los Alamitos (1995)Google Scholar
  20. 20.
    van Raamsdonk, F.: Confluence and superdevelopments. In: Kirchner, C. (ed.) RTA 1993. LNCS, vol. 690, pp. 168–182. Springer, Heidelberg (1993)Google Scholar
  21. 21.
    Tait, W.W.: Intensional interpretation of functionals of finite type I. J. Symbolic Logic 32, 198–212 (1967)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Zurab Khasidashvili
    • 1
  • Adolfo Piperno
    • 2
  1. 1.Department of Mathematics and Computer ScienceBen-Gurion University of the NegevBeer-ShevaIsrael
  2. 2.Dipartimento di Scienze dell’InformazioneUniversità di Roma “La Sapienza”RomaItaly

Personalised recommendations